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How to Place Points and Fractions on a Unit Circle

To work with trigonometry functions, it’s helpful to be able to place points and fractions on a unit circle. The unit circle is a circle with its center at the origin of the coordinate plane and with a radius of 1 unit.

Any circle with its center at the origin has the equation x2 + y2 = r2, where r is the radius of the circle. In the case of a unit circle, the equation is x2 + y2 = 1. This equation shows that the points lying on the unit circle have to have coordinates (x- and y-values) that, when you square each of them and then add those values together, equal 1.

The four intercepts of the unit circle.
The four intercepts of the unit circle.

The coordinates for the points lying on the unit circle that intersect the two axes are (1,0), (–1,0), (0,1), and (0,–1). These four points (called intercepts) are shown in the preceding figure.

The rest of the points on the unit circle aren’t as nice and neat as those you see in the preceding figure. They all have fractions or radicals — or both — in them. For instance, the point

image1.png

lies on the unit circle. Look at how these coordinates work in the equation of the unit circle:

image2.png

If you square each coordinate and add those values together, you get 1.

Any combination of these two coordinates, whether the coordinates are positive or negative, gives you a different point on the unit circle. They all work because, whether a number is positive or negative, its square is the same positive number. Here are some combinations of those two coordinates that satisfy the unit-circle equation:

image3.png

Another pair of coordinates that works on the unit circle is

image4.png

because the sum of the squares is equal to 1:

image5.png

The numbers that continually crop up as coordinates of points on the unit circle are

image6.png

They should look familiar — they’re the sine and cosine values of the most common acute-angle measures. The following figure shows the locations of those points on the unit circle.

Points on the unit circle.
Points on the unit circle.

The points on the unit circle shown in the preceding figure are frequently used in trigonometry and other math applications, but they aren’t the only points on that circle. Every circle has an infinite number of points with all sorts of interesting coordinates — even more interesting than those already shown. If you’re looking for the coordinates of some other point on the unit circle, you can just pick some number between –1 and 1 to be the x- or the y-value and then solve for the other value. All these other coordinates come into play when you’re drawing a ray that starts at the unit circle’s center and want to find the trig functions of the angle formed by that ray and the positive x-axis.

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